Advertisement

The Laplacian with Boundary Conditions

  • David Borthwick
Chapter
  • 288 Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 284)

Abstract

The abstract theory developed in previous chapters is applied to the spectral theory of the Laplacian on a bounded open set in Euclidean space. We define the sefl-adjoint extensions corresponding to classical boundary conditions. The chapter includes discussion of some classic spectral results, including the Weyl law, Courant’s nodal domain theorem, and eigenvalue comparison theorems.

References

  1. 1.
    Arendt, W., Nittka, R., Peter, W., Steiner, F.: Weyl’s law: spectral properties of the Laplacian in mathematics and physics. In: Mathematical Analysis of Evolution, Information, and Complexity, pp. 1–71. Wiley-VCH, Weinheim (2009)Google Scholar
  2. 2.
    Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36, 235–249 (1957)Google Scholar
  3. 4.
    Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Spectral Theory and Geometry (Edinburgh, 1998). London Mathematical Society Lecture Notes Series, vol. 273, pp. 95–139. Cambridge University Press, Cambridge (1999)Google Scholar
  4. 5.
    Ashbaugh, M.S., Benguria, R.D.: Proof of the Payne-Pólya-Weinberger conjecture. Bull. Am. Math. Soc. 25, 19–29 (1991)CrossRefGoogle Scholar
  5. 13.
    Borthwick, D.: Introduction to Partial Differential Equations. Universitext. Springer, Cham (2016)CrossRefGoogle Scholar
  6. 18.
    Carleman, T.: Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes. In: Åttonde Skandinaviska Matematikerkongressen (Stockholm, 1934), pp. 34–44. Håkan Ohlssons Boktryckeri, Lund (1935)Google Scholar
  7. 19.
    Chavel, I.: Eigenvalues in Riemannian Geometry. Academic, London (1984). Including a chapter by Randol, B, With an appendix by Dodziuk, J.Google Scholar
  8. 20.
    Chavel, I.: Isoperimetric Inequalities. Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press, Cambridge (2001)Google Scholar
  9. 24.
    Davies, E.B.: Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, vol. 42. Cambridge University Press, Cambridge (1995)Google Scholar
  10. 29.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 2nd edn., vol. 19. American Mathematical Society, Providence (2010)Google Scholar
  11. 34.
    Gårding, L.: On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators. Math. Scand. 1, 237–255 (1953)MathSciNetCrossRefGoogle Scholar
  12. 36.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1983)Google Scholar
  13. 38.
    Gordon, C., Webb, D., Wolpert, S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110, 1–22 (1992)MathSciNetCrossRefGoogle Scholar
  14. 46.
    Ivrii, V.: 100 years of Weyl’s law. Bull. Math. Sci. 6, 379–452 (2016)MathSciNetCrossRefGoogle Scholar
  15. 48.
    Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73, 1–23 (1966)MathSciNetCrossRefGoogle Scholar
  16. 56.
    Lamé, G.: Mémoire sur la propagation de la chaleur dans les polyèdres. J. Ecol. Polytech. 22, 194–251 (1833)Google Scholar
  17. 59.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  18. 64.
    Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V.: NIST Digital Library of Mathematical Functions (2016). http://dlmf.nist.gov/. Release 1.0.19
  19. 65.
    Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)MathSciNetCrossRefGoogle Scholar
  20. 67.
    Pleijel, Å.: A study of certain Green’s functions with applications in the theory of vibrating membranes. Ark. Mat. 2, 553–569 (1954)MathSciNetCrossRefGoogle Scholar
  21. 68.
    Pólya, G., Szegő, G.: Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, vol. 27. Princeton University Press, Princeton (1951)Google Scholar
  22. 70.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic, London (1975)Google Scholar
  23. 77.
    Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill, New York (1976)Google Scholar
  24. 80.
    Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)Google Scholar
  25. 88.
    Szegő, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3, 343–356 (1954)Google Scholar
  26. 95.
    Weinberger, H.F.: An isoperimetric inequality for the N-dimensional free membrane problem. J. Rational Mech. Anal. 5, 633–636 (1956)MathSciNetzbMATHGoogle Scholar
  27. 97.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • David Borthwick
    • 1
  1. 1.Department of MathematicsEmory UniversityAtlantaUSA

Personalised recommendations